TITLE

1. Suppression of Magnetic Islands In Stellarator equilibrium calculations By iterating the plasma equilibrium equations to find the plasma equilibrium, and adjusting the coil geometry at each iteration to cancel the resonant fields normal to the rational rotational-transform surfaces, magnetic islands are suppressed in stellarator equilibria. Engineering requirements are satisfied and the plasma is stable to ideal MHD modes. The method is applied with success to the National Compact Stellarator eXperiment. TITLE Presented by Stuart R. Hudson for the NCSX design team Columbia University Plasma Physics Colloquium 9/20/2002

2. The NCSX Team

3. Outline • Coil-healing is required because stellarators will in general have islands. Healed fixed-boundary equilibria may be constructed, but coil design algorithms cannot reproduce a given boundary exactly. • The coil-healing algorithm is presented. • Features of coil-healing are discussed in detail : • construction of rational surfaces, calculation of resonant fields, • expressing resonant fields (and additional constraints) as function of coil-geometry, • adjustment of coils to remove resonances. 4) Coil-healing results for NCSX are presented showing a stable stellarator equilibrium, with build-able coils, with “good-flux-surfaces”.

4. Resonant fields cause magnetic islands perturbation shear islands + =

5. Magnetic field line flow is a Hamiltonian system • May write • From which • Field line flow • Field line flow determined by Field Line Hamiltonian 

6. Structure of field examined with Poincaré plot. toroidal geometry Poincaré surface of section magnetic field lines reduce continuous system to discrete mapping type of field line Poincaré plot irrational on flux surface closed curve rational on flux surface discrete points stochastic / chaotic finite volume

7. Islands and chaos caused by perturbation integrable increasing perturbation chaotic

8. Islands may be removed by boundary variation. • Equilibrium (including islands & resonant fields) is determined by plasma boundary. m=5 island suppressed large m=5 island Equilibrium before boundary variation Equilibrium after boundary variation to remove islands

9. Coil healing is required because … • Plasma and coil design optimization routines rely on equilibrium calculations. • All fast equilibrium codes (in particular VMEC) presuppose perfect nested magnetic surfaces. • Existence / size of magnetic islands cannot be addressed. • Practical restrictions ensure the coil field cannot balance exactly the plasma field at every point on a given boundary. Instead … • The spectrum of B.nat the boundary relevant to island formation must be suppressed. • Coil alteration must not degrade previously performed plasma optimization (ideal stability, quasi-axisymmetry,…). • Coil alteration must not violate engineering constraints.

10. The equilibrium calculation and coil healing proceed simultaneously via an iterative approach. Based on the PIES code A single PIES/healing iteration is shown 1) Bn = BPn + BC(n) total field = plasma + coil field 2) p = J(n+1)  Bn calculate plasma current 3) J(n+1) =   BP calculate plasma field 4) BP(n+1) =BPn+(1- )BP blending for numerical stability 5) B = BP(n+1) + BC(n) nearly integrable field resonant fields function of  6) –Bi= BCij  nj coil geometry adjusted 7)j(n+1) = jn + jn At each iteration, the coil geometry is adjusted to cancel the resonant fields n iteration index ; BP plasma field ;BCcoil field ;  coil geometry harmonics ; =0.99 blending parameter ; BCij coupling matrix.

11. After the plasma field is updated, a nearly integrable magnetic field is identified. 5) B = BP(n+1) + BC(n) • The total magnetic field is the sum of the updated plasma field, and the previous coil field, and may be considered as a nearly integrable field. • Magnetic islands can exist at rational rotational-transform flux surfaces of a nearby integrable field, if the resonant normal field is non-zero. • NOTE : The coil geometry is adjusted after the plasma field is updated, and the plasma field is not changed until after the coil geometry adjustment is complete.

12. Rational surfaces of a nearby integrable field are constructed. • Quadratic-flux minimizing surfaces may be thought of as rational rotational transform flux surfaces of a nearby integrable field. Dewar, Hudson & Price.Physics Letters A, 194:49, 1994Hudson & Dewar, Physics of Plasmas 6(5):1532,1999. 2) Quadratic-flux surface functional 2 is a function of arbitrary surface  : • Extremizing surfaces are comprised of a family of periodic curves, • integral curves of , along which the action gradient  is constant. • Such curves may be used as rational field lines of a nearby integrable • field.

13. The field normal to the rational surface is calculated. illustration of quadratic-flux-minimizing surface Poincare plot on =0 plane (red dots) B= BP+ BC() n e e For given BP, the resonant normal field is a function of coil geometry Cross section (black line) passes through island =0 plane

14. Engineering constraints are calculated. • To be “build-able”, the coils must satisfy engineering requirements. • Engineering constraints are calculated by the COILOPT code. • In this application, the coil-coil separation and coil minimum radius of curvature are considered. single filament description of coils radius of curvature : must exceed iR0 coil-coil separation : must exceed iCC0 Coil-coil separation and minimum radius of curvature expressed as functions of coil geometry

15. Plasma stability is calculated. For given coil set,… free-boundary VMEC determines equilibrium, TERPSICHORE / COBRA give kink / ballooning stability Kink stability and ballooning stability expressed as functions of coil geometry

16. A dependent function vector is constructed. 1) The selected quantities form a constraint vector B : resonant fields coil-coil separation radius of curvature kink, ballooning eigenvalue

17. A matrix coupling constraint vector Bto coil geometry is defined. • The coils lie on a winding surface with toroidal variation given by a set of geometry parameters : • The dependent vector Bis a function of a chosen set of harmonics  • First order expansion for small changes in  : • is calculated from finite differences. Coupling Matrix of partial derivatives of resonant coil field at rational surface

18. The coils are adjusted to cancel resonant fields. • A multi-dimensional Newton method solves for the coil correction to cancel the resonant fields at the rational surface modular coils plasma new coil location winding surface

19. Application to NCSX. The method is applied to the design of the proposed National Compact Stellarator Experiment

20. 5.5-m 3.4 The NCSX coils are shown.

21. The NCSX modular coil geometry will be adjusted.

22. The healed configuration has good-flux-surfaces. VMEC initialization boundary =3/5 surface Poincare plot on up-down symmetric =2/6 =3/6 surface high order islands not considered.

23. The healed configuration is greatly improved compared to the original configuration. healed coils unhealed coils large (5,3) island Though islands may re-appear as profiles &  vary, the healed coils display better flux surface quality than the original coils over a variety of states. chaotic field

24. Healed states enable PIES-VMEC benchmarks red curve : PIES flux surface dashed curve : healed coils VMEC  if the PIES equilibrium were perfectly ‘healed’, the PIES boundary and the VMEC boundary should agree. • the VMEC boundary for the original coils is the solid line; •  the VMEC boundary for the healedcoils is the dashed line; •  a PIES flux surface for the healedcoils is drawn as the red line; • the agreement is good, but needs to be quantified; • discrepancies may result from high-order residual islands; • numerical convergence tests should be performed.

25. Healed PIES / VMEC comparison healed VMEC original VMEC PIES High order islands and ‘near-resonant’ deformation may explain the discrepancy between the healed PIES and VMEC boundary (dashed) for the healed coils.

26. The magnitude of the coil change is acceptable. • plot shows coils on toroidal winding surface • coil change  2cm • coil change exceeds • construction tolerances • does not impact machine design (diagnostic, NBI access still ok) • * the resonant harmonics have been adjusted healed original

27. Finite thickness coils show further improvement. Inner wall Single filament equilibrium Multi filament equilibrium The multi filament coils show further improvement

28. The healed coils support good vacuum states. The healed coils maintain good vacuum states

29. Robustness of healed coils • The discharge scenario is a sequence of equilibria, with increasing , that evolves the current profile in time self-consistently from the discharge initiation to the high  state. • The healed coils show improved configurations for this sequence. • NOTE : this sequence has in no way been optimized for surface quality.

30. Healed coils are improved for discharge seq. with trim coils original coils

31. Trim Coils are included in the NCSX design.

32. Trim Coils provide additional island control without trim coils: (n,m)=(3,6) island with trim coils: island suppressed equilibrium from discharge scenario

33. Coil-Healing : Summary and Future Work • The plasma and coils converge simultaneously to a free-boundary equilibrium with selected islands suppressed, while preserving engineering constraints and plasma stability. • Adjusting the coil geometry at every PIES iteration enables effective control of non-linearity of the plasma response to changes in the external field. • In the limit of suppressing additional, higher order islands, this approach can lead to high-pressure integrable configurations. • Extensions to the method include : • including constraints on rotational transform, location of magnetic axis and boundary; optimizing quasi-symmetry, . . • speed improvements by parallelization, improved numerical techniques, . . • simultaneously `healing’ multiple configurations (eg. various ’s, . . )